The Efficiency of Bargaining with Many Items

The Efficiency of Bargaining with Many Items
Matthew O. Jackson, Hugo F. Sonnenschein, and Yiqing Xing∗
March 2015
Abstract
For an important class of alternating-offer bargaining problems with significant
two-sided asymmetric information, we demonstrate that all sequential equilibria are
efficient or approximately efficient. We focus on the case in which the bargaining
problem contains many aspects, the value of each of which is private information,
but which aggregate to an overall surplus from trade that is approximately known.
The results may help explain why so many situations with significant asymmetric
information exhibit little departure from first best efficiency (e.g., time lost to labor
negotiations is negligible world-wide).
Keywords: Bargaining, Exchange, Trade, Multiple Objects, Linking, Contract Theory, Mechanism Design, Implementation, Bayesian Equilibrium, Efficiency, Sequential
Equilibrium, Trembling.
JEL Classification Numbers: A13, C72, C78, D47, D82.
∗
Department of Economics, Stanford University, Stanford, California 94305-6072 USA, and Department
of Economics, University of Chicago, Chicago, Illinois, 60657. Jackson is also an external faculty member
at the Santa Fe Institute and a fellow of CIFAR. Emails: [email protected], [email protected]
and [email protected]. Financial support from the NSF under grant SES-1155302 as well as ARO MURI
award W911NF-12-1-0509. We thank seminar participants at the University of Chicago, Princeton University,
Stanford University and the Decentralization conference, for helpful comments.
1
Introduction
It is not curious that strikes exist, but it is curious that they are so rare. According to
Kennan (2005), in the United States between 1948 and 2005, “idleness due to strikes never
exceeded one half of one percent of total working days in any year.”1 In fact, since 1990
average lost time has been about 20 minutes per year per worker in the U.S.; and, even in
a more strike-prone country like Spain, the number is less than 1/3 of a day per worker per
year (again, according to Kennan (2005)).
The puzzle here has its roots in the two seminal papers in bargaining: Rubinstein (1982)
and Myerson and Satterthwaite (1983). On the one hand, if there is complete information
about the value of a potential bargain and any impatience among bargainers, then Rubinstein
showed that in a canonical and descriptively attractive alternating offer model of bargaining
there is always immediate agreement and efficiency in a unique equilibrium outcome. While
that is encouraging, it is far from the explanation of the data quoted above, as it is likely
that there is some private information in many settings in which firms and workers bargain
over labor contracts, and in bargaining settings more generally. In that light, Myerson and
Satterthwaite’s important result comes to bear which shows that with two-sided uncertainty
about the value of trade, it can be that all equilibria result in substantial inefficiency.2
In this paper we provide an explanation for why it can be that bargaining with substantial
two-sided uncertainty can still systematically result in efficient outcomes. We analyze settings
in which there may be several or many aspects to the bargain,3 each of which may have
significant uncertainty, but on average aggregate to a fairly predictable surplus. For example,
there may be substantial uncertainty to how much workers value a health package, time
off for child care, pension provisions, sick days, flexible schedule, safety rules, disability
insurance, grievance provisions, wages, and so forth; and similarly how management views
the relative costs of each of these items. If each item was bargained over in isolation, following
1
Kennan notes similar numbers for Canada, where he mentions that lost work time was about a third
of a day per year per worker. Kennan also states that “Although the data are not readily available for
a broad sample of developed countries, the pattern described above seems quite general: days lost due to
strikes amount to only a fraction of a day per worker per annum, on average, exceeding one day only in a
few exceptional years.”
2
See, e.g. Kennan and Wilson (1993), Ausubel, Cramton and Deneckere (2001) for surveys on bargaining
with private information. Later results include Deneckere and Liang (2006) for the case of interdependent
information; and Feinberg and Skrzypacz (2005) for the case of higher-order uncertainty.
3
For the literature on exchanges with multiple objects: Armstrong (1999) considers monopolistic price
discrimination problems with multiple products, and Sen (2000) and Inderst (2003) consider bargaining
over both price and quantity (of homogeneous goods). All those models have one-sided private information
about buyers valuation. In contrast, our paper consider the case where agents on both sides hold private
information on heterogeneous products, so incentive issues potentially are substantial on both sides and
become impediment to efficiency.
1
Myerson and Satterthwaite (1983) there would be substantial losses in efficiency. However,
in aggregate, both the workers and the firm may have a good idea of the overall cost or value
that an equilibrium total compensation package should have. As we prove, when bargaining
over many aspects, or many items, at once, it is possible to reach efficient outcomes (in all
equilibria) even though this would not happen if the aspects or items were each bargained
over in complete isolation.
Thus, the settings we examine live between the extremes considered by Rubinstein (1982)
and Myerson and Satterthwaite (1983): each item has substantial uncertainty, but the overall
surplus is (nearly) known. A priori, it is not clear which of the extreme reasonings should
apply, if either. We show that which logic applies depends on the properties of the bargaining
protocol: If the protocol allows the agents to bargain over more than one item at a time
then efficiency ensues and for all equilibria, while if the protocol has agents bargaining over
the items independently, then none of equilibria are inefficient, even though offers could be
coordinated across items.4 Thus, the contrast across bargaining protocols is quite stark. The
critical insight is that to achieve efficiency the responder cannot select some items and not
others at various prices, but must face coordinated prices across items.
A reaction to our results might be that we already knew the answer to one of the questions
posed here, namely that efficiency is possible. In particular, in an earlier work (Jackson and
Sonnenschein (2007)), we showed that in settings in which collective decisions over many objects could be linked together, there exist mechanisms that result in (approximately) efficient
outcomes. There is an important difference however. Those familiar with that paper will
recall that it takes a mechanism design approach to the problem of efficient allocation and
makes substantial use of a ‘mechanism designer’ or ‘mediator’ who is well-informed regarding the nature of the Bayesian uncertainty. Here we consider a more ambitious conclusion
in which the mechanism is not designed, but is a natural bargaining protocol. We examine the circumstances under which bargaining competition necessarily leads to an efficient
allocation of resources (think of the relative infrequency of strikes and efficient contractual
arrangements) when there are many items to be bargained over with significant uncertainty
regarding their valuations, but relatively good knowledge regarding the overall surplus. We
consider extensions of the Rubinstein-Stahl bargaining protocol to the case where the bargain has many items. We refer to these extensions as alternating-offer bargaining protocols.
As in Rubinstein-Stahl, the bargaining agents have similar strategy spaces and they make
offers in turn. Offers involve demands of surplus together with constraints on what is made
available to the recipient if she accepts the offer. Bargaining ends when the process of offers, counter-offers, and eventual agreement followed by transfers, has run its course. In the
4
If offers cannot be coordinated across items, then the Myerson-Satterthwaite result applies directly and
in many cases guarantees significant departure from efficiency. See subsection 4.2 for more discussion.
2
spirit of the word “bargaining” we consider protocols that “impose” relatively little and are
not customized to account for the individuals who may meet to bargain or their particular
uncertainty. We have in mind the essence of the dialogue in a labor-management negotiation.
Although the alternating-offer bargaining protocols that we consider can be viewed as
“mechanisms,”5 they describe interactions between agents who come together to bargain
and are not a priori restricted. In particular, the protocols allow us to investigate when it
is that the competitive forces of bargaining, which arise naturally in a world where agents
can claim to have any valuations and ask for any share of a hypothetical surplus, will result
in efficiency. Our modeling leads to the conclusion that for the important class of problems
under consideration the forces of bargaining competition are sufficient for efficiency. Furthermore, since our description of bargaining competition does not vary with the structure
of uncertainty in the world, we provide a new angle on the “robust mechanisms” literature.
In particular, the bargaining protocols that we examine do not depend on any information
about the distribution of agents’ types or the possible surplus, hence can work across many
different environments.6 Some very natural bargaining protocols provide for a sort of ‘universal’ implementation, in that they result in efficient outcomes in all equilibria for a general
class of settings.
2
Multi-Object Bargaining
We first introduce the multi-object bargaining setting.
2.1
Multiple Objects and Preferences
A multi-object bargaining problem consists of:
• a deal with n objects (or aspects, items),
5
There is an extensive mechanism design literature characterizing the feasibility of efficient allocations.
See, e.g., Jackson (2003) and Segal and Whinston (2012) and the references therein for static case; and
Skrzypacz and Toikka (2014) for dynamic case.
6
This follows in the broader spirit of Wilson’s (1987) criticisms of mechanisms that depend on agents’
(higher-order) beliefs. Satterthwaite, Williams, and Zachariadis (2014) also view such mechanisms as “impractical” as “[the agents’] beliefs are not a datum that is practically available for defining economic institutions” (p.249). In contrast, “robust” and “detail-free” have mainly been used in literature to refer to
resolving the more explicit aspect of the Wilson’s critique, namely the assumption of common knowledge
among agents: e.g., see Bergemann and Morris (2005) and Roughgarden and Talgam-Cohen (2013). The
exception is Matsushima (2008) who used “detail-free” with a meaning more similar to ours in an auction environment. To avoid confusions in terminologies we use “universality” to capture the feature that a
protocol/mechanism is not defined upon the knowledge of the prior distribution.
3
• two agents, {s, b}, the seller s and the buyer b,
• finite valuation or type spaces Θi ⊂ IR, i ∈ {s, b}, with Θ = (Θs )n × (Θb )n , and
• a probability distribution described by a density function f over types Θ.
• We let fi denote the marginal of f on Θi .
We assume throughout that there exist (θs , θb ) ∈ Θs × Θb such that θb > θs , so that there
are, at least potentially, nontrivial gains from trade.
In the beginning of period 0, the types are drawn according to the density f and agents
observe their own types only. The realized types are denoted θi = (θi1 , . . . , θik , . . . θin ), with
θik being agent i’s value for aspect/object k. Bargaining occurs at t = 0, 1, 2, . . .
The generality of f allows for correlated values and also allows for different distributions
over various classes of objects (say some big, some small). We discuss conditions on f under
which efficient bargaining holds with our main results.
The bargainers are considered to be engaged in making a single joint decision or deal
that generalizes Rubinstein-Stahl in that it has several aspects, while in Rubinstein-Stahl,
modified to allow for Bayesian uncertainty, surplus is divided between a buyer and a seller
when the good is transferred and dollars are paid. Here two firms may be bargaining on a
contract which includes specification of a service to be provided, or a good to be produced,
and this could include many aspects which need to be specified, which include the time to
production, various specs of the object (e.g., its weight, performance, durability), the quality
of the object, penalties for failure to deliver. Or for example it could be a contract between
a faculty member and a university, specifying a teaching load, a sabbatical policy, research
funding, summer support, a salary, etc. In another interpretation there is a buyer and a
seller with the seller being able to deliver any of the n objects to the buyer. The allocation
problem is then which objects will be transferred and how much money will pass from the
buyer to the seller.7 With this noted, we use the terminology of “objects” or “items”, with
the reader keeping in mind that this all applies to multiple aspects of a single exchange.
2.2
Payoffs
Time advances in discrete periods and is discounted according to δi ∈ (0, 1), i ∈ {s, b}.
We also assume that the agents’ payoffs across objects are additively separable, but this
is not essential to the analysis (see subsection 4.3 for more discussions).
7
The case in which some objects are initially held by one agent and other objects by the other could be
easily accommodated, but we keep the notation uncluttered.
4
If Xt is the set of objects traded at date t, and pt is a value (total price) paid from the
buyer to the seller in period t, then the seller’s net (realized) utility from trade as seen from
time 0 is
!
X
X
θsk
δst pt −
Us0 =
t
k∈Xt
and the utility to the buyer is
!
Ub0 =
X
t
δbt
X
θbk
!
− pt .
k∈Xt
Note that this specification allows different objects to trade at different times (via the
Xt ’s). A special case is in which trade of all goods must occur in only one period (so for
instance, disability insurance might or might not be included and then the price would
implicitly include how much of it would be paid by the employer).
The discounting can be interpreted in at least two (standard) ways. 1) Production occurs
at time t, so the seller incurs a cost θsk for item k at time t, and the buyer also gets the payoff
from consuming at t. This interpretation also applies to the example of strikes between a
firm and a union, where the costs and benefits are held until the employment relationship
is restored (, i.e. the agreement is reached in the bargaining, at t). 2) The seller holds the
items each of which generates a flow payoff in every period up to period t, when she forgoes
the future flow payoffs for those traded items, i.e. θsk is the time-t value of flow payoffs the
seller could get from item k.
2.3
2.3.1
Benchmark Protocols
Rubinstein-Stahl Bargaining with Many Objects: A Minimalist Extension
As an important benchmark, we specify a natural extension of the Rubinstein-Stahl bargaining protocol to multiple objects.
• One of the agents, say the seller, announces a string of her valuations θbs = (θbs1 , . . . , θbsn )
and demands a surplus vs ∈ V , from some compact set V of possible total values.
• The other agent, say the buyer, accepts or rejects.
If accepted, the buyer picks a subset of items to buy, X ⊂ {1, . . . , n}, X 6= ∅, and
delivers a net payoff equal to vs based on the seller’s announced values: so, the
total price given to the seller by the buyer is equal to
X
vs +
θbsk .
k∈X
5
In the case in which it is the buyer who has announced θbb = (θbb1 , . . . , θbbn ) and
demanded a net payoff vb ∈ IR+ , then the seller picks the X and the price paid
by the buyer is
X
θbbk − vb .
k∈X
The game ends.
• If rejected we start again with the roles of the agents reversed (and one period of
discounting ensues).
2.3.2
The Item-by-Item Bargaining Protocol
Another natural benchmark protocol is one in which agents simultaneously bargain over the
n objects, but the bargaining is independent across items - so each item is bargained upon
via its own price and sequence of offers and counter-offers. It is still possible that agents may
tie the bargaining together across items, but only via their equilibrium actions, as otherwise
it is simply n separate bargaining processes that are conducted in parallel.
• One of the agents, say the seller, announces a string of prices for items ps = (ps1 , . . . , psn ).
• The other agent, say the buyer, can accept any subset of the items, which are then
traded at those prices.
• If any objects are untraded, we start again with the roles of the agents reversed (and
one period of discounting ensues). The buyer then announces a price for each item
that still remains untraded.
• The seller can accept any subset of those remaining items, which are then traded at
the buyer’s suggested prices.
• If any objects are untraded, we start again with the roles reversed. We continue in this
manner indefinitely or until all goods are traded.
• Agents obtain net utility discounted to period 0 utiles for the value of any trades
relative to the period in which a particular object was traded.
We note that in some cases, with bargaining over a contract with many aspects, this
sort of bargaining may raise issues of feasibility and interpretation if agents cannot consume
until all of the objects are agreed upon. For instance, an employment contract would have
to specify wages, pension plan, hours, holidays, etc., and employment might not be feasible
until all of the facets were agreed upon. In that case, it might be that only the previous
6
protocol would be feasible and this item-by-item protocol might not. In contrast, a consumer
thinking about buying several carpets, might bargain with a seller on a item-by-item basis
as in this protocol, or for a set of carpets as in the previous protocol.
2.3.3
A Combinatorial Bargaining Protocol
One could also move to the opposite extreme: instead of bargaining item-by-item, people
could bargain in ways that allow them to consider trading all possible subsets.
Agents alternate in offering.
• One of the agents, say the seller, names a total price for each possible subset of the
objects. Formally, an offer is a mapping p : 2n → R+ .
• The other agent, say the buyer, accepts or rejects.
• If accepted, the buyer picks items to trade, X ⊂ {1, . . . , n}, and the buyer pays the
price p(X) to the seller.
• If rejected we start again with the roles of the agents reversed (and one period of
discounting ensues).
The richness in message space of the combinatorial protocol is a disadvantage in practice:
the size of an offer is 2n , exploding at exponential rate as n grows. For instance, with n = 20
each offer needs to specify 220 ≈ 1048k prices, which is not very realistic in practice, where
agents tend to use “reduced forms” like the one introduced in Section 2.3.1 where a much
smaller message spaces suffices to convey the most important information. In Section 2.3.4
we discuss another natural protocol with small message spaces.
2.3.4
A Frequency Protocol as a ‘Reduced-Form’ Protocol
To go to another extreme, we examine a protocol in which agents use minimal announcements
(of frequencies of types and demanded values), until an agreement is reached and then
announce specific values for actual objects in a second stage. This fits with many settings
in which people bargain over “basic terms”, and then after they have reached a tentative
agreement then they fill in the details.
Consider the following “frequency protocol” constituting of two phases:
Phase 1:
The offerer, say the seller quotes a frequency distribution φbns ∈ Φns ⊂ ∆Θns ,8 and
a vs ∈ V .
8
Φni is the set of possible frequencies (with n items) on Θns . Notice that the quoted frequency φbni may
different from i’s true frequency.
7
The recipient accepts or rejects.
If accepted we move to Phase 2.
If rejected we begin Phase 1 again with the roles of the agents reversed (and one
period of discounting ensues).
Phase 2:
(We do this as if the seller made the offer and the buyer accepted in Phase 1,
there is an obvious change for the other case)
The seller announces θbsn ∈ Θns that has a frequency distribution φbns .
Either the buyer picks a subset of items to buy X ⊂ {1, ..., n}, and makes a
P
n
payment of at least vs + k∈X θbsk
to the seller for those objects; or the buyer says
“No” and there is no trade.
The game ends.
Note that all of the protocols introduced in Sections 2.3.1- 2.3.4 are “universal”, in the
sense that the protocols are defined independently of the prior distribution f of the players’
valuations. This also applies to the frequency protocol as the feasible frequencies that one
can announce do not depend on the prior distribution. In contrast, the “linking mechanisms”
introduced in Jackson and Sonnenschein (2007) were defined relative to the distribution of
types (see Section 2.3.6). Universal protocols have some advantages in practice as they can
become ‘traditions’ or ‘customs’ for how to bargain and apply across many settings, rather
than requiring adjustment to fit details of the setting.
2.3.5
A General Definition of an Alternating-Offer Bargaining Protocol
With these various examples of alternating-offer bargaining protocols in hand, we provide a
definition of what we mean in general by an alternating-offer bargaining protocol when there
are n objects.
Given a finite set A, let Π(A) denote the set of all partitions of A, with a generic partition
denoted π.
• One of the agents, i (the offerer), announces from a finite set of possible announcements
(‘offers’) A0i , with a generic offer denoted a0i .
• The other agent, j (the recipient), observes some information about the offer I(a0i )
which is the element of a partition π 0 ∈ Π(A0i ) that contains a0i . That agent then
responds from a set A0j (I(a0i )), with a generic response denoted by a0j .
8
• As a function of a0i , a0j some goods (possibly the empty set), denoted X(a0i , a0j ) ⊂
{1, . . . , n}, are traded, and some price is paid from the buyer to the seller, and those
goods are consumed. The set of goods traded and the price are commonly observable.
• One period of discounting ensues.
• Inductively, in period t, agent i(t) makes offers from a finite set Ai(t) (ht−1 ) which could
t−1
depend on ht−1 ≡ (a0i(0) , a0j(0) , . . . , at−1
i(t−1) , aj(t−1) ), the full history of bargaining through
the last period9
• The other agent j(t) reacts from a set Aj(t) (ht−1 , I(ati(t) )), where I(ati(t) ) is the element
of a partition π t ∈ Π(Ati ) that contains ati and is the information that j observes about
i’s latest offer.
• As a function of ht = (ht−1 , ati(t) , atj(t) ), some of the remaining goods are traded and
consumed.
• This continues as long as there are goods remaining to be traded.
The information observed, I(ati(t) ), is shaped by the protocol: I(ati(t) ) = {ati(t) } if the
offerer’s action is perfectly observed by the responder, I(ati(t) ) = Ati(t) if the offerer’s action
is not observable. Non-observability allows us to consider simultaneous announcement protocols within the same framework. The frequency protocol can be fit into this framework by
having the offerer announce both the frequency, demand, and type profile simultaneously,
but then the responder only sees the frequency and not the type profile.
To see how our first protocol from Section 2.3.1 fits into this definition, note that it is
easy let it be such that once any goods are traded, then the remaining announcements are all
irrelevant and no further trade is possible and so the game effectively ends. The item-by-item
protocol is one in which the set of items changes over time, but otherwise the mechanism
looks ‘stationary’ in terms of the announcements.
2.3.6
A Linking Mechanism
This protocol is based on the mechanisms from Jackson and Sonnenschein (2007).
• Agents simultaneously announce θbi ∈ Θni with the restriction that their announced
9
For instance, only some objects might remain, or a protocol might not allow an agent to change prices
in certain directions, etc. This specification also allows for various rules for choosing the proposer - an easy
extension is to allow for random recognition of i(t).
9
frequency of valuations must match the empirical distribution fi ;10 i.e.,
|{k|θbik = θ}| = fi (θ)
for each θ ∈ Θi .
• The outcome is efficient trade at some pre-determined prices relative to the announced
θbb , θbs .
So, trade of a particular object occurs if and only if θbb > θbs . In the case in which the
distribution of the types of each agent is known and exchangeable, any(!) function p(θsi , θbi )
will actually result in an incentive compatible mechanism (including, for instance, the Rubinstein prices) - which follows from Theorem 1 in Jackson and Sonnenschein (2007). Notice
this mechanism does not have “universality” since the knowledge of the prior distribution is
essential to design the mechanism.
3
Challenges in Multi-Object Bargaining
In order to illustrate some of the challenges and main ideas, we consider a particular example.
In the example, the seller’s valuations lie in Θs = {0, 8}, and the buyer’s valuations lie
in Θb = {2, 10}.
To start, consider an f such that the marginal distribution to any particular object k is
such that it puts equal likelihood on each combination of (θsk , θbk ) ∈ {(0, 2), (0, 10), (8, 2), (8, 10)}.
If the bargaining were just on one object, then we know that there is no Bayesian incentive
compatible, Pareto efficient, and ex post individually rational direct mechanism for this
problem, and thus there is no alternating offer bargaining protocol with the aforementioned
properties. This follows in a general form from Myerson and Satterthwaite (1983), but can
be established directly for this setting.
Now consider bargaining with multiple objects, and for illustration purpose suppose
the agents have valuations independently drawn across objects. As the number of objects
becomes large, although uncertainty about the total surplus may vanish, it is still unclear
whether we should expect that the Rubinstein-Stahl extension from Section 2.3.1 or the
item-by-item protocol from Section 2.3.2 will lead to nearly efficient outcomes.
There are two forces that push in different directions regarding whether we get efficiency
as we have many objects for trade:
10
For ease of exposition, we suppose that this is possible. See Jackson and Sonnenschein (2007) for details
of how to extend the mechanism in the case in which the integer counts of an announced distribution cannot
match up with the expected frequencies exactly.
10
• There is less uncertainty about overall/per item surplus.
• There is more uncertainty about which items to trade and at which prices.
Thus, there are several issues here with such alternating-offer bargaining protocols: (i)
Does reducing uncertainty about overall surplus lead to nearly efficient trade? and (ii) When
there are multiple objects can the issue of deciding over “which objects to trade at which
prices” be solved? and (iii) How do the answers to these questions depend on whether we
are talking about some versus all equilibria?
First note that the challenge is due to the combination of incentives and individual rationality constraints that arise with such bargaining protocols: they naturally imply (interim)
individual rationality, since agents bargain (and need to accept an offer) after they observe
their own valuations.11 A successful elicitation of trade-relevant information needs to subsidize an agent differently enough depending on the agent’s valuations; on the other hand,
interim individual rationality pins down how much an agent should receive with her worst
possible type. The two together determine the least amount of share each agent should
receive, and efficiency is impossible when the total expected surplus is not big enough to
deliver both agents their shares.
Let us discuss some of these challenges in turn, beginning with the limiting case in which
overall surplus is commonly known. For that case, we show that all sequential equilibria
of the the protocol from Section 2.3.1 lead to immediate and efficient exchange, despite the
substantial uncertainty about which objects to trade. Moreover, we show that the same is
true for a general class of bargaining protocols.
4
Multi-Object Bargaining with Known Surplus
We focus first on a case in which the surplus is commonly known, but which objects should
trade and at which prices is not known. We think of this as the natural limit of worlds
in which there are a large number of aspects. (One might think of a parallel between
11
It is important to note that eliminating individual rationality allows for an efficient and (dominant
strategy!) incentive compatible mechanism for this problem: The price is always 5, and the agents announce
their types and trade occurs unless the announced types are (8,2) in which case no trade occurs. It is easily
seen that this mechanism has truth as a dominant strategy and it is efficient, but it violates the individual
rationality of the agents who sometimes ends up with negative values. In fact, the mechanism fails to be
interim individually rational for either agent even simply conditional upon type (a type 8 seller only ever
gets a price of 5, and a type 2 buyer sometimes pays 5 but never less) - although it is ex ante individually
rational. (See Gresik 1991 for results on the extension of Myerson and Satterthwaite’s results to more general
distributions and for interim individual rationality, and noting the possibility of finding ex ante individually
rational mechanisms for some distributions.)
11
the assumption of commonly-known surplus with only a finite number of objects and the
assumption that finite numbers of agents in Walrasian economies take a common price vector
as given - concentrating on the insights of how such economies work and then later checking
that the results still hold when one drops the assumption and looks at limits of sequences of
economies.) We discuss the extensions to the case of a “nearly known” surplus in Section 5.
In particular, a bargaining problem (n, Θ, f ) (as defined above) has a known surplus
S¯ > 0 if there exists S¯ > 0 such that
1 X
max(θbk − θsk , 0) = S¯
S(θs , θb ) ≡
n k=1,...,n
for every (θs , θb ) for which f (θs , θb ) > 0.
Thus, a bargaining problem with known surplus is such that agents know what the total
gains from trade are, but not necessarily which objects should be traded.
A first question is whether in this case one ends up with results that are closer to Rubinstein’s conclusion of efficiency since overall surplus is known, or Myerson and Satterthwaite’s
inefficiency results since there can still be arbitrary uncertainty about whether it is efficient
to trade any particular object.
4.1
An Efficiency Result
We first consider the natural extension of the Rubinstein-Stahl protocol that we introduced
in Subsection 2.3.1. As we will see in Theorem 1, despite the substantial uncertainty about
the value of any given object, all sequential equilibria of that protocol lead to to immediate
and efficient trade and a unique division of the total surplus.
Theorem 1 If a bargaining problem (n, Θ, f ) has a known surplus S¯ > 0, then in all sequential equilibria under the protocol introduced in subsection 2.3.1:
• the agreement is reached immediately,
• the full surplus is realized, and
• agents’ expected net payoffs from trade are their Rubinstein shares, i.e.
seller, and
δb (1−δs )S¯
1−δb δs
(1−δb )S¯
1−δb δs
for the
for the buyer.
All proofs appear in the appendix.
We emphasize that this protocol is ‘universal’ in that the above result above holds for
exactly the same protocol for any f with a known surplus. This distinguishes the result from
a mechanism-design result in which the mechanism is tailored to the f . Thus, this is not only
distinguished because we are taking a positive perspective (examining a mechanism which
12
seems ‘natural’ in terms of how people actually bargain) as opposed to a normative one
(using direct mechanisms to prove that efficiency is possible if the ‘designer’ has sufficient
knowledge), but also because the protocol in question is both simple and directly adapts
with the environment.
The intuition behind Theorem 1 is as follows. If there were any inefficiency in the
anticipated equilibrium path, then since the agents know the potential surplus and can
make demands for shares of that total surplus, there is an offer that they each know makes
them strictly better off if it is immediately accepted. The existence of such an offer rules
out inefficient equilibria. The argument for the precise Rubinstein shares is based on an
extension of that by Shaked and Sutton (1984).
4.2
The Bargaining Protocol Matters
One may conjecture that with known overall surplus, the above result about efficient trade
would extend to most any bargaining protocol. This is not the case. The uncertainty
about specific matchups can distort the agents’ incentives and lead to substantial delay and
inefficiency in protocols in which agents cannot make offers that require trading on all objects
at once.
To illustrate this point, we consider the item-by-item bargaining protocol in which agents
bargain on objects separately but simultaneously as defined in Section 2.3.2.
Under this protocol, not only do there exist equilibria with substantial inefficiency, but
in many situations substantial inefficiency occurs in all sequential equilibria. This is shown
in the following example.
Example 1 Consider agents bargaining over n = 2 objects under the item-by-item bargaining protocol. The seller has type (θs1 , θs2 ) = (0, 0), and there are two equally likely type
profiles for the buyer: (θb1 , θb2 ) = (2, 10) or (10, 2). In all sequential equilibria at most one
of the two objects is traded in the initial period.
The proof behind the claim in the example appears in the Appendix.
Out of situations with uncertainty, this example is such that it should be easiest to reach
efficiency since it is commonly known that all objects should trade and there is only onesided uncertainty about whether trade should occur.12 Nevertheless, even in such a simple
environment, none of the sequential equilibria are efficient with item-by-item bargaining.
This result is in contrast to Theorem 1, in which all sequential sequential equilibria are
efficient under the protocol from Section 2.3.1.
12
For reviews of single-object bargaining with one-sided uncertainty, see, e.g. Fudenberg, Levine and Tirole
(1985) and Ausubel, Cramton, and Deneckere (2002).
13
The inefficiency in Example 1 derives from the seller’s incentives to screen the buyer’s
type to try to obtain a high price for the more valuable item. Such incentives are mitigated
when agents can bargain on items “overall”, as agents have better knowledge about the
overall surplus. Instead, in protocols that entail separate bargaining, even though agents can
coordinate their actions across objects they cannot take advantage of the greater knowledge
that they have about the overall values of trade by offers that involve overall surplus, and
the ensuing selection problems are not easily overcome.
In the above example, the inefficiency is necessarily large when discounting is substantial,
but that leaves open the possibility that, for δ very close to 1, approximate efficiency could
also be achieved via the item-by-item protocol. Indeed, extending results from Ausubel and
Deneckere (1989) to this multi-object problem, it becomes clear there exist a large set of
equilibria only some of which would be approximately efficient in this example. Nonetheless,
this example illustrates that the item-by-item protocol performs much differently from protocols in which objects are bargained over together, and that substantial inefficiencies are
possible and sometimes ubiquitous when items are bargained over separately, but completely
precluded when objects are bargained over simultaneously.
4.3
A General Feature for a Protocol to Provide Efficiency
The bargaining protocol 2.3.1 allows the agents to bargain over the surplus and discover
which objects to trade in an integrated manner that takes advantage of their knowledge
of the overall surplus, while item-by-item bargaining does not. There is a general sense in
which any protocol that has such integration leads to full efficiency (in all equilibria), as we
now formalize.
¯
Consider a bargaining problem (n, Θ, f ) with a known surplus S.
Definition 1 (Share-demanding offers and protocols) A protocol includes a share¯ and after a history
v demanding offer in some period t, for some i(t), θi(t) ∈ Θni and v ∈ [0, S]
ht−1 if:13
• there exists ai ∈ Ai(t) (ht−1 ) such that
for every aj ∈ Aj(t) (ht−1 , I(ati(t) )) either there is no trade or the realized surplus
for i(t) in the current period is at least v, and
for any θj(t) for which f (θi(t) , θj(t) ) > 0: there exists aj ∈ Aj(t) (ht−1 , I(ati(t) )) for
which the realized surplus in the current period for i(t) is v and the realized surplus
in the current period is S¯ − v for θj(t) .
13
Payoffs expressed here are not-discounted; i.e., they are evaluated in the current period.
14
¯ if at any
A bargaining protocol includes share-demanding offers for some set V ⊂ [0, S],
point of the protocol through which no trade has yet occurred, the current offerer i(t) has a
share-v demanding offer for each v ∈ V and type θi(t) for which fi(t) (θi(t) ) > 0.
When there is no ambiguity, we say a bargaining protocol “includes share-demanding
¯
offers” if it includes share-demanding offers for [0, S].
We say that a bargaining protocol is “fully alternating” if the offerer’s action is always
observable to the recipient, i.e. I(ati(t) ) = {ati(t) }.
Most of the protocols that we have discussed are fully alternating. An exception is
the frequency protocol as it involves subperiods so that only frequency announcements are
observable in the first phase. Nonetheless, the result below extends to it, but requires a
special proof. Since that proof is a special case of Theorem 3 we omit it.
Theorem 2 If a bargaining problem with n objects has a known surplus S¯ > 0 and the
protocol includes share-demanding offers and is fully alternating, then in all sequential equilibria:
• the agreement is reached immediately,
• the full surplus is realized, and
• the agents’ expected payoffs equal to their Rubinstein shares; i.e.,
and
δb (1−δs )S¯
1−δb δs
(1−δb )S¯
1−δb δs
for the seller,
for the buyer.
The protocol introduced in 2.3.1 includes share-demanding offers: at any point of the
game and for any share v, a share-v demanding offer is (θi , v), i.e. the current offerer lists
the valuations truthfully, and demands a total share of v. Such an offer, once accepted, gives
the offerer exactly payoff of v regardless of the recipient’s choice of the objects, and gives
the recipient S¯ − v if the recipient chooses to trade all objects with values higher than the
costs.
On the other hand, the item-by-item bargaining protocol does not include share-demanding
offers: the offerer’s payoff depends on which objects the recipient accepts to trade - and the
offerer cannot ask for an overall surplus that must be taken as a whole rather than in part.
This creates a tension between the offerer’s incentives and the realization of full surplus. For
instance, in Example 1 the seller is able to demand a value of v = 6, for instance by asking
for a price of 3 on each object. However, the buyer would rather just accept the price of 3
on the more valuable object, thus realizing a total surplus of 7, instead of accepting both
and only getting a surplus of 6. This would only lead to a surplus of 3 for the seller.
Besides the protocol 2.3.1, many other protocols include share-demanding offers. For
example, the combinatorial protocol (from Section 2.3.3) includes share-demanding offers.
15
In addition, note that any offer available in the protocol introduced in Subsection 2.3.1 has
an equivalent offer in the combinatorial protocol.14 As a result, the combinatorial protocol
can be viewed as a “richer” protocol in terms of the message space. Moreover, that protocol
has the advantage of allowing for a general payoff structure. In particular, the agents’ payoffs
can be non-additively separable across objects, but quasi-linear in money; i.e.
vs = p − us (X, θs ),
vb = ub (X, θs ) − p,
S(θs , θb ) = maxX (ub (X, θb ) − us (X, θs )),
where θi ∈ Θni is agent i’s type. Again time is discounted according to δi ∈ (0, 1). Notice
that the above are total payoffs and surplus.
The frequency protocol (from Section 2.3.4) also includes share demanding offers, and
even though it is not a ‘fully alternating’ protocol an argument similar to that underlying
Theorem 2 shows that its equilibria also result in efficient outcomes.15 The simplicity of
frequency protocol’s strategy space is also helpful when we turn to settings with incomplete
information about the surplus.
Again we note that the protocols just mentioned all satisfy our “universality” condition;
i.e., the protocols work for any distribution and surplus.
4.4
Reopened Bargaining for Not-Yet-Traded Objects
Our efficiency results discussed above are robust to reopening the bargaining over the notyet-traded objects. In particular:
• In the protocol 2.3.1, after the recipient accepts an offer, selects which items to trade,
and pays a total price (that delivers the surplus demanded by the offerer), the agents
may continue to bargain over the remaining objects, with the last recipient being the
first offerer in the continuation - with the same protocol used on the remaining items.
14
P
Any offer (e.g. from the seller) (θs , vs ) in the protocol 2.3.1 has an equivalent offer p(X) = k∈X θsk +
vs , ∀X in the combinatorial protocol.
15
This holds when type distributions are exchangeable; a sufficient condition for which is that the valuations
are i.i.d. across items and independent across players. Then in all exchangeable sequential equilibria, where
agent adopts the same strategy (in Phase 1) for types that have the same frequency, Phase 2 has a unique
equilibrium: the recipient chooses those items with positive value-cost gaps, provided the sum of those gaps
exceeds the payoff demanded by the offerer; and as a best response, the offerer’s lists the valuations as
truthful as possible - in particular, the offerer is truthful when the truthful frequency is announced in Phase
1 (which is the case in the share demanding offers).
16
• In the combinatorial protocol 2.3.3, after the recipient accepts an offer, picks a subset
and pays the price for that subset, the agents may continue to bargain over the remaining objects, with the last recipient being the first offerer in the continuation according
to the same combinatorial protocol.
In both cases, the agents receive the payoffs from the objects traded (and the price paid)
at the time when the objects are traded. This implies that different discounting applies to
objects traded in different periods.
Both protocols with reopening still include share-demanding offers. Take the protocol
2.3.1 as an example: being truthful and demanding a total surplus of nv always guarantees
nv to self once the offer is accepted, regardless of which objects are not picked up and how
they are traded when the bargaining reopens; on the other hand, when the recipient accepts
such an offer, the best strategy is to choose to trade all objects with the buyer’s values higher
than the seller’s costs so that no objects with positive surplus are left not traded.
Therefore, as a corollary to Theorem 2, it follows that the efficiency of all sequential
equilibria is robust to the opportunity of reopening the bargaining.
The opportunity to reopen exchange complicates analyses of many settings, from Walrasian exchange, to auctions, to contracting. Here, the robustness comes from the sharedemanding offers and the knowledge of the overall surplus. When there is substantial uncertainty, the potential to reopen discussions distorts incentives for screening, which are
completely circumvented with the share-demanding offers.
5
Multi-Object Bargaining with a ‘Nearly-Known’ Surplus
Our analysis with a known total surplus illustrates the idea that with known surplus, sharedemanding offers provide for efficient bargaining and are really essential for such results.
The ability to bargain over a full bundle, means that the known surplus dominates over the
screening of particular items. Thus, we end up with a sort of “Rubinstein” result, rather
than a “Myerson-Satterthwaite” result, with many items and uncertainty over each item but
a known surplus overall.
The exact knowledge of the full surplus is a expository device, as we can imagine that
with large numbers of objects agents will have a good idea of the total gains from trade, but
still have substantial uncertainty about which precise set of objects should trade. Thus, it is
useful to verify that there is not a substantial discontinuity between having the total surplus
being ‘nearly-known’ versus exactly-known. And, given that all equilibria are efficient in the
limit, it is enough to look for upper-hemi continuity.
17
In this section we explore that continuity. There are several technical difficulties with
establishing upper-hemi continuity.
The first is that incomplete information game theory is still not well-understood in the
case of continua of types and actions - as measurability issues and other issues of updating
beliefs conditional on atomless events is cumbersome (e.g., sequential equilibria have only
even been recently defined for such settings, and existence is still a tricky problem). This
prompts us to work with finite environments and protocols - so that measurability issues are
avoided.
The second difficulty is that in order to have the total surplus be ‘nearly-known’ it makes
sense to work large numbers of objects - so that one can appeal to laws of large numbers.
Unfortunately, that also means that the action space in our previously discussed bargaining
protocols explodes exponentially. This leads to challenges in characterizing how beliefs must
evolve in equilibria.
The third difficulty is that the updating of beliefs is problematic even in very simple
incomplete information games - and upper-hemi continuity can require some restrictions.
To handle these issues we work with four different approaches:
• We analyze the frequency protocol (which limits the size of the strategies and makes
it easier to identify beliefs);
• We work with a fixed number of objects with uncertainty that converges to full knowledge, and have agents tremble so that beliefs are tied down;
• We work with large number of objects but place bounds on the rate at which agents
update beliefs over time; and
• We analyze other protocols in which the strategy space satisfies a size restriction, but
still allows for value-demanding offers.
So, in this section we begin by discussing the setting and illustrating some of the technical
issues via some simple examples. We then turn to each of the four approaches mentioned
above.
5.1
Multi-Object Bargaining with Converging Surplus
We index things by a sequence m, to allow (but not require) the rate at which uncertainty
converges not to depend directly on the number of objects. The m-th economy has nm
objects.
A sequence of bargaining problems with priors f m ∈ ∆(Θns m × Θnb m ) have surpluses
converging to S¯ > 0 if
¯ as m → ∞
S m →p S,
18
where S m is the per-item surplus in the m-th problem; i.e., a random variable S(θs , θb ) ≡
P
1
m
k=1,...,nm (θbk − θsk )+ with types (θs , θb ) drawn according to f .
nm
This embodies the idea that there can be substantial uncertainty about which objects
should trade and at which prices. In addition, the above allows for correlations of valuations
across objects and between agents. It also allows for heterogeneities across objects.
5.2
A Technical Challenge: Failure of Upper-Hemicontiunity of
Sequential Equilibria at the Limit of Certainty
We first illustrate the challenge that is substantial but is of a ‘technical’ nature: sequential
equilibria often fail a fundamental upper hemi-continuity condition. We view this as a
shortcoming of sequential equilibrium and the current tool-box of game theory. Moreover,
this is not solved by standard existing refinements.
A game with arbitrarily small (Bayesian) uncertainty is very different from its counterpart
with certainty, in the sense that some sequences of sequential equilibria of the former game
have no limit in the set of sequential equilibria (subgame-perfect equilibria) of the limit
game with certainty. This happens because the the notion of sequential equilibria allows for
a lot of freedom in off-path beliefs, and as a result lots of outcomes can be supported as
part of a sequential equilibrium by extreme off-path beliefs, and leads to a failure of basic
conditions like upper hemicontinuity of the equilibrium correspondence (which generally
holds for Bayesian equilibrium: see, e.g. Jackson, Simon, Swinkels and Zame (2002), but
fails for sequential equilibria). This challenge is not specific to our multi-object bargaining
games. It applies to many simple games, and in particular, upper hemi-continuity even fails
in simple single-object Rubinstein bargaining with the most basic forms of uncertainty.16
Consider alternating-offer bargaining with one object and one-sided uncertainty (let δs =
δb = δ = 0.8): the buyer’s value is commonly known as 10, and the seller’s cost is either
0 or 8, so that it is commonly known that the agents should always trade immediately to
get efficiency. In addition, suppose the seller’s cost has increasing probability on 8 along
the sequence. One may conjecture that all sequential equilibria in this game converge to
16
The problem that we are pointing out here is endemic: the example still works with perturbations
in the payoffs and/or how the small uncertainty is introduced, as it is freedom in specifying beliefs that
cause problems, and not exact indifferences (which lead to lower hemi-continuity problems). Thus, there
is a fundamental sort of discontinuity between equilibrium concepts with slight amounts of incomplete
information and the limit of full information, which seems symptomatic of the tools of game theory rather
than a real phenomenon. This does not contradict the fact that when both the sequence of priors and its limit
are in the interior of the distribution space, the set of sequential equilibria satisfies upper hemi-continuity
(Kreps and Wilson (1982), Proposition 2, p.876). Here upper-hemi continuity fails since we are converging
to complete information. Given the importance of the complete information case in the theory and (its
approximation) in practice, the failure of upper hemi-continuity is important and disturbing.
19
the unique equilibrium in the limiting complete information bargaining game with seller’s
cost being 8 for sure, where the unique equilibrium outcome is immediate trade at a price
2
p8,10 = 1+δ
+ 8 ≈ 9.11. However, this is not the case. With arbitrarily small uncertainty so
that fs (8) = 1 − ε for any tiny ε , sequential equilibria allow for substantial inefficiency, and
a wide range of prices at which the agents trade. We illustrate this point with the following
example:
Example 2 With above parameters, there exists a sequential equilibrium with no trade in
the first period. In particular, the following occurs on equilibrium path: at t = 0, both types
2δ
+ 8 ≈ 8.89,
of the seller offer p = 10 and are rejected; at t = 1, the buyer offers a p = 1+δ
the Rubinstein price with cost 8 and value 10, accepted by both types of the seller.
To see why the claim is true, consider the following equilibrium specification. The onpath behavior is supported by the buyer’s off-path belief Pr(θs = 8) = 0 upon seeing any
offer p 6= 10 at t = 0.17 Given this belief, the buyer plays as if in a complete information
10δ
Rubinstein bargaining with with “0 meets 10”, i.e. always offering p = 1+δ
≈ 4.44, and
rejects any offer with p > 5.56. It is then easy to verify that given the buyer’s off-path
behavior, both types of the seller prefers to stay on path.
The problem of substantial inefficiency as presented in Example 2 remains with a discount
factor arbitrarily close to 1: one can construct sequential equilibria with no trade in the first
several periods, and that the efficiency loss from delay is at least as big as in the example
(1 − 0.8 = 20%).
With arbitrarily small uncertainty, the set of sequential equilibria allows for substantial
multiplicity in outcomes, unlike the uniqueness feature in the limiting complete information
game. This multiplicity is due to dramatic belief updatings off-path: in Example 2, upon
seeing one off-path offer, the buyer believes the seller is of a 0 type for sure, i.e. almost
completely discards the prior belief which puts almost all weights on the 8 type. This
dramatic change in beliefs results in a continuation of the game that is very different from the
one where the agents started with, hence a lack of unique prediction of outcomes. Whereas
in the limiting complete information game, there is no room for belief updating. Hence
there is generally no upper hemi-continuity of the set of sequential equilibria at the limit of
certainty.
Given the freedom of belief updating, it is not surprising that many outcomes can occur
in an equilibrium.
17
It is direct to check that this satisfies the consistency conditions of sequential equilibrium, as one can
have a sequence of mixed strategies where the 0 types are arbitrarily more likely to play strategies other
than 10 compared to the 8 types.
20
However, as motivated before, we aim to understand the case in which the overall surplus
is increasingly known. In addition, we aim to study properties of all equilibria. Therefore,
imposing additional structure on the game is necessary due to deal with above challenge.
Our main approach is to introduce trembles (which can be small, but are not forced to 0)
to deal with belief-updating - this places all actions on the equilibrium path and precludes
the problems due to lack of restrictions, and avoids us having to make ad hoc restrictions,
concerning off-path beliefs. Alternative approaches and results are discussed in Appendix
A.
5.3
Approximate Efficiency Results with Trembles
In this part we work with bounded grids of prices and introduce trembles that naturally
regulate the rate at which beliefs can be updated. We first illustrate the idea in the singleobject bargaining game.
5.3.1
Single-object bargaining.
First, reconsider the example from Section 2.
Now consider some grid of prices, so that P ∆ is finite with a grid {0, ∆, 2∆, . . . , Smax },
18
where Smax = max θb − min θs .
Consider a variation on the game such that there is some small 0 < γ < 1 such that at
every node in the game, each type of each player who moves at that node places probability
at least γ/|P ∆ | on each possible offer (or γ/2 on each of accept/reject for the recipient), but
subject to that constraint places the remaining probability according to a best response to
the agent’s beliefs.19 So, it is as if an agent best responds with probability 1 − γ and then
trembles with the remaining probability γ - picking an action uniformly at random. The
exact uniform at random aspect of trembles is not needed, as is clear from the proofs; but it
is necessary that the trembles not become infinitely more likely to some actions than others.
With trembles, all nodes are reached and so beliefs are completely tied-down by Bayes’ Rule,
and so we can work with a trembling version of Bayesian equilibrium in which agents’ update
18
The increment ∆ can be viewed as a smallest currency unit (e.g., van Damme, Selten, and Winter
(1990)).
19
Our trembles relate to the equilibrium refinement literature (e.g., Kreps and Wilson (1982), Rubinstein
(1985), Banks and Sobel (1987), Grossman and Perry (1986), and Cho and Kreps (1987).) In particular,
Kreps and Wilson (1982) also use trembles when defining sequential equilibria, but consider a sequence of
vanishing trembles, so the size of trembles become eventually negligible, whereas we consider a limit theorem
where the size of trembles is fixed (although they can be arbitrarily small) and then there is vanishing
uncertainty about overall surplus. Our motivation is quite different from the literature on bargaining with
“reputational types, where each agent has some type(s) being fully rational and some being irrational (e.g.,
Compte and Jehiel (2002), Abreu and Pearce (2007), Wolitzky, (2012), and the papers cited therein.)
21
beliefs via Bayes’ rule at all nodes and they best respond with probability 1 − γ and tremble
with the remaining probability.
Example 3 Consider P ∆ = {0, 1, 2, ..., 10}, and γ = 0.11 so that the probability of trembles
to each possible price (in all periods) is γ/|P ∆ | = .01. If the prior is .999 on some type, then
the posterior after one-period of belief updating is at least .9 on that type.
The claim in this example follows easily from bounds on Bayesian updating (see Lemma 1
in the Appendix). In particular, 1 − Pr(ai | θi0 ) ≤ (.01)−1 × .999 ≤ .1, hence Pr(ai | θi0 ) ≥ .9.
When agents tremble at some minimal rate, after one period of belief updating the
posterior still puts substantial probability on the most probable types. Therefore, even very
small trembles can make the posteriors more congruent with the priors in each period.
The following proposition shows that this convergence of beliefs is enough to generally
restore continuity of the equilibrium correspondence with respect to small amounts of uncertainty.
Proposition 1 Consider a single-object alternating-offer bargaining game. For any ε > 0
and correspondingly fine enough P ∆ , there is a small enough tremble probability γ(ε) > 0,
such that if γ ∈ (γ(ε), 0) then there exists low enough uncertainty (f γ < 1 such that f places
probability at least f γ on a single pair of types (θs , θb )) such that in all Bayesian equilibria,
with probability at least 1 − ε:
• if θb > θs , then the price offered in the initial period is in ((1 − ε)p, (1 + ε)p) and is
b )(θ b −θ s )
accepted, where the price p = θs + (1−δ1−δ
is the Rubinstein price associated with
b δs
the high probability types θs , θb ; and
• if θb < θs , then trade does not occur.
Thus, Proposition 1 shows that introducing small trembles can completely eradicate the
discontinuities associated with incomplete information. The proof of this proposition is a
variation on that of Theorem 3, and so we omit it.
Let us be explicit about the order of the quantifiers, as the order is subtle, but not as
restrictive as it might superficially appear. Given any ε it is clear that we need a fine enough
grid and small enough trembles to be sure that an ε-approximate efficiency is possible. If the
grid is too coarse then the right prices could not be chosen, and if the trembles are too likely
then bargaining breakdown because of random behavior becomes too likely. Once these are
fine enough so as not to get in the way of efficiency, they can be as small as we like. However,
as trembles become smaller, we need closer to complete information, f γ close to 1, so that
belief on ‘very unlikely types’ cannot become too large under trembles.
22
Effectively, trembles tie down beliefs and avoid the problems of updating off the path
that drove the discontinuities in sequential equilibria without sustained trembles. Here, we
get a continuity result at the limit (a technique that could also be helpful in other settings,
beyond bargaining).
5.3.2
Multi-object bargaining under the frequency protocol.
Next, we extend the analysis of approximate efficiency with trembles to multi-object bargaining, under the frequency protocol introduced in Section 2.3.4. We work with the efficiency
protocol since it has a “small” strategy space and this makes the handling of beliefs under
trembles tractable. We discuss extensions to other protocols in the appendix.
Let V n = {0, 1, 2, ..., nSmax } be the grid of total surplus from which an agent can demand.20
Again, consider trembles in Phase 1 in any period by all types of any player with probability γ, uniformly to each of the feasible actions.21 As noted above, uniformity of the
trembles is an expository convenience, and all that is needed is that trembles are distributed
in a manner such that the relative probability of trembling to any two different actions is
bounded above (and hence below).
The following distributional assumptions help put the frequency protocol to work. For
simplicity, suppose agent i’s (i = s, b) valuations (θi1 , ..., θik , ..., θin ) are i.i.d distributed
according to a frequency f¯i over Θi , and independent across agents.22 Without loss of
generality, let f¯i (θik ) > 0, ∀θik ∈ Θi (defining Θi to be the support).
With such distributions, there is an expected surplus (per object) of
S¯ =
XX
θs
f¯s (θs )f¯b (θb ) max{θb − θs , 0}.
θb
With these exchangeable distributions, it is natural to restrict attentions to exchangeable
strategies and equilibria, i.e. an agent adopts the same strategy (in Phase 1) for types that
have the same frequency. We can then prove the following approximate efficiency result for
the frequency protocol.
20
One can view this as a grid of pennies (or dollars, etc.), and can be as fine as possible simply by
renormalizing Smax .
21
Trembles are not needed in Phase 2. Nonetheless, Theorem 3 is robust to adding similar trembles to
Phase 2 as well.
22
These assumptions are stronger than needed. All that is needed is that the distribution over types is
exchangeable (f m remains the same under any permutation of θ: if π is a bijection, then f m (θπ ) = f m (θ) for
all θ, where θkπ = θπ(k) ) and 2) exponential rate of precision improvements, sufficient conditions for which are
stationary and with summable covariance (cf. https://stat.duke.edu/courses/Fall11/sta205/lec/wk-07.pdf
section7.2.)
23
Theorem 3 Consider a sequence of problems indexed by the number of objects n with bargaining under frequency protocol with given δs , δb , and i.i.d. distributions over types on each
object according to some f¯s , f¯b . For any ε > 0, there exist a small enough tremble probability
γ(ε) > 0, such that for any γ ∈ (0, γ(ε)) there exists nγ such that if n > nγ then:
1. There exist Bayesian equilibria with trembles that are exchangeable.23
2. In any such equilibrium, with probability at least 1 − ε:
• agreement is reached in the initial period,
¯ and
• the realized surplus is at least (1 − ε)S;
• Expected payoff / ‘Full-Information Rubinstein share for the realized types’ for
each agent lies in (1 − ε, 1 + ε).
Theorem 3 is a limit theorem, showing that our “at the limit” Theorem 1 is robust to introducing uncertainty, once one addresses technical challenges associated with discontinuities
in equilibrium concepts.
Although we do not provide rates of convergence, they are direct to deduce for Theorem
3. The inefficiency per-item, when trembles are set to be as small as possible as a function of n (as otherwise they drive the inefficiency), is of order O(n−0.5−τ ), ∀τ > 0. Thus,
the inefficiency vanishes at a rate arbitrarily close to the square-root of n. In particular,
inefficiency comes from several sources: First, there is a potential inefficiency due to the
increment in price grids. Here that naturally disappears with n since bargaining is over total
surplus which is normalized by the number of items, so the inefficiency due to the lumpiness
in the bargaining grid is of the order O( n1 ). Second, there is inefficiency due to the trembles,
which is proportional to γ, which can be picked as disappearing with n, in particular of
order O( n1 ). Third, the realized surplus can be different from the limit surplus, and such a
difference induces inefficiency that is of the order of O(n−0.5−τ ), ∀τ > 0, a rate similar to
ones provided by standard central limit theorems .24
23
In this game with trembles all nodes are reached by all types with positive probability, and so perfect
Bayesian equilibria (and hence sequential equilibria) and Bayesian equilibria coincide.
24
Fixing any rate of trembles γ > 0, and time T such that the time-0 continuation value after period T is
negligible, the rate at which time-T posterior on some event may different from time-0 prior is of the order
O(γ −T n|Θi |T ) due to the trembles, where the size of i’s action space is ∼ O(n|Θi | ). Setting γ ∼ O( n1 ), this
becomes O(n(|Θi |+1)T ). Let d be an amount allowed between the realized and limit surpluses. By standard
concentration inequalities, e.g. Hoeffding (1963), the beliefs that the actual surplus and realized surplus
2
differs by more than d (at some belief condition on a period-T ) is at most αT ∼ O(e−2d n × n(|Θi |+1)T ) which
2
is still of the order O(e−2d n ) since the exponential term dominates.
2
max{O(d), O(e−2d n )} is minimized with an optimal selected difference d ∼ O(n−0.5−τ ), resulting in an
overall inefficiency of the order of O(n−0.5−τ ). Notice that the square-root rate of convergence cannot be
achieved because we need to simultaneously control the difference allowed and the likelihood of the tails.
24
The theorem is stated for exchangeable equilibria. We suspect that the result also holds
for non-exchangeable equilibria, but in those cases the second phase of the protocol becomes
more difficult to analyze, as now an agent may have a posterior that places more weight on
some types with a given frequency than others. We conjecture that a similar result holds
when extending to those equilibria, as the rate at which helpful information is gained is
bounded by the trembles, but we have not been able to find a proof or counter-example for
the non-exchangeable case.
6
Concluding Remarks
Despite the fact that actual bargaining frequently involves the need to get to agreement
on several aspects of a contract or deal, the formal theory of alternating offer bargaining,
as exemplified by the seminal work of Rubinstein (1982) and the literature that follows,
focuses on a situation in which there is a single aspect. We extend that theory to encompass
bargaining when there are many aspects to a contract and Bayesian uncertainty regarding
the valuation of each of these aspects, yet no uncertainty about overall surplus. We analyze
alternating offer bargaining protocols as natural extensions of Rubinstein (1982). These
protocols are ‘universal’ or ‘detail-free’, in the sense that the protocols are not tailored to
the uncertainty of the setting. We show that all “share-demanding” protocols (a rich and
natural set of protocols) lead to efficient exchange in all sequential equilibria, while objectby-object protocols result in inefficiency for all sequential equilibria.
In showing that our results are robust to small amounts of uncertainty about the overall surplus, we confronted a fundamental game-theoretic difficulty: the failure of upperhemicontiunity of sequential equilibria at the limit of certainty. Our approach of introducing non-vanishing trembles may be useful as a natural approach to restoring upper hemicontinuity of the equilibrium correspondence in general settings - a topic for further research.
The investigation of mechanisms that are independent of the uncertainty faced by the
agents, and operate for many settings, is an important issue for applications beyond bargaining and auctions.
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Appendix
A
Proofs
Proof of Theorem 1:
This Theorem is a corollary to Theorem 2 given the fact that the protocol 2.3.1 includes
share-demanding offers. In particular, for any v, a share-v demanding offer is to announce
the truth θin and demand a payoff of v.
Proof of Example 1:
Let Ls be the seller’s worst continuation payoff in any seller-offer period in any sequential
equilibria with both objects remain. This means when the buyer makes an offer, he gets a
continuation payoff of at most δ(12 − δLs ) since the seller can always rejects on both objects
and counteroffers in the subsequent period.
Consider the seller’s offer (p, p) with some p > 2. The buyer rejects p on the value-2
object, and accepts p on the value-10 object for sure if p < p˜, s.t.
10 − p˜ +
2δ
= (12 − δLs )δ,
1+δ
28
2δ
where on the left-hand side 10 − p is the payoff from the value-10 object and 1+δ
from the
value-2 object (the corresponding Rubinstein share, since it is commonly known that the
object left is of value-2).
Therefore, with an offer of (˜
p − , p˜ − ) ∀ > 0, the seller can always get an acceptance on
2δ 2
− .
the value-10 and a discounted Rubinstein share on the value-2 , i.e. a payoff of p˜ + 1+δ
On the other hand, since Ls is the seller’s payoff in some SE, it must exceeds the payoff
from the above deviation (˜
p − , p˜ − ). This requires
2δ 2
− .
Ls ≥ p˜ +
1+δ
This, combined with the definition of p˜, gives (1 − δ 2 )Ls ≥ 10(1 − δ) − , i.e. (since can
be arbitrarily small)
10
Ls ≥
1+δ
Finally, for both objects to be traded in the initial period the seller’s expected payoff is
at most 4, which is not possible in any sequential equilibria: In order to have both objects
traded with a positive probability, the seller’s strategy in the first period must put positive
weight (if mixing) on an offer that has prices at most 2 on each object. The seller gets a payoff
of at most 4 from such an offer, and hence an expected payoff of at most 4 from the game
since the seller must be indifferent among any strategies used with positive probability.
Proof of Theorem 2:
We begin with some notation. Let Θn (f ) ≡ {(θs , θb ) | f (θs , θb ) > 0} be the set of (profiles
of) types that are “possible” under the joint prior distribution f , and Θni (f ) ≡ {θi | fi (θi ) >
0} is similarly defined for agent i.
Note that the assumption of known surplus implies that S(θs , θb ) = S¯ for all (θs , θb ) ∈
Θn (f ). In addition, in a sequential equilibrium, after any history, the joint posterior distribution f˜’s support is a subset of Θn (f ), and similarly the posterior over i’s type has a support
as a subset of Θni (f ). This is true both on and off the equilibrium path since consistent
beliefs must have a support that is a subset of the prior’s support.
Let Mit be the supremum of the expected per-item continuation payoff for agent i, starting
at the beginning of period t over all sequential equilibria histories, and all i’s types in
θ˜i ∈ Θni (f )
We now establish the upper and lower bounds of the seller’s utility in any equilibrium, as
well as the buyer’s utility, and show that they all correspond to a unique equilibrium payoff
that corresponds to immediate and efficient trade, and the Rubinstein shares.
¯
We first show that M 0 ≤ 1−δb S.
s
1+δs δb
29
At t + 1 (k even), the buyer makes the offers. We argue that any buyer with θb ∈ Θnb (f )
can guarantee a payoff arbitrarily close to
Lt+1
≡ S¯ − δs Mst+2 .
b
The buyer does so by offering a share-vt+1 demanding offer with vt+1 ≡ n(Lt+1
− η) for
b
η > 0 arbitrarily small. Such an offer is accepted for sure for a seller with any type θbs s.t.
¯ therefore by
(θbs , θb ) ∈ Θn (f ): notice that θbs ∈ Θns (f ) by construction, hence S(θbs , θb ) = S;
accepting the offer (and picking any items on which her cost is lower than the announced
buyer values) the seller gets δs Mst+2 + η, which exceeds δs Mst+2 , the present value of the
payoff from the continuation of the game if rejecting the offer. Finally, the buyer gets a
payoff of Lt+1
− η if the above offer is accepted, regardless which items the seller picks to
b
trade.
At t, the seller makes offer. We argue that a seller with any type θs ∈ Θns (f ) can get a
n
payoff at most S¯ − δb Lt+1
b : With any type θb ∈ Θb (f ), by rejecting an offer at t, the buyer’s
payoff from the continuation of the game has a present value of at least δb Lt+1
− δb e for
b
t+1
n
¯
∀η > 0. Hence the payoff left to the seller with θs ∈ Θs (f ) is at most S − δb Lb , as the
(expected) surplus is S¯ by construction.
≤ (1 − δb )S¯ + δs δb Mst+2
By definition of Mst , we have Mst ≤ S¯ − δb Lt+1
b
The above is true for any k = 0, 2, 4, ...,. Iteratively applying the above leads to
Ms0 ≤
1 − δb ¯
S.
1 + δs δb
By a similar argument, it follows that
L0s ≥
1 − δb ¯
S.
1 + δs δb
Therefore, the payoff for the seller with any possible type θs ∈ Θns (f ) in any sequential
1−δb ¯
equilibrium is Us ≤ 1+δ
S.
s δb
1−δb ¯
1−δs ¯
From the above, we also know that Ms2 = L2s = 1+δ
S, hence Mb1 = L1b = 1+δ
S
s δb
s δb
1−δb ¯
(both in terms of the present value then), i.e. the total surplus realized is at least 1+δs δb S +
1−δs ¯
δb 1+δ
S = S¯ which is the surplus from efficient trade. Hence the bargaining outcome must
s δb
be efficient, which means immediate trade with the efficient set of items being exchanged.
The utility terms correspond to the Rubinstein shares.
Next, we turn to the case with uncertainty. We begin by a lemma that establishes a rate
of updating in a protocol with trembles.
Lemma 1 For any event E ⊂ Θi , let Pr(E) be its prior in some period and Pr(E | ai ) be
the posterior one-period after conditional an action ai . It follows that
Pr(E | ai ) ≤ Pr(E)/γ,
30
where γ > 0 is the lower bound of the size of trembles (from any type) to ai .
Proof of Lemma 1:
Giving updating according to Bayes’ rule:
Pr(E | ai ) =
Pr(ai | E) Pr(E)
≤ Pr(E)/γ,
Pr(ai | E) Pr(E) + Pr(ai | E c ) Pr(E c )
where E c is the complement of E, and the inequality comes from Pr(ai | E) ≤ 1 and
Pr(ai | ·) ≥ γ due to trembles.
Proof of Theorem 3:
Notation:
Let Φni ⊂ ∆(Θi ) be the collection of all possible frequencies of n items with valuations
picked from Θi .
Let φ[θin ] ∈ ∆(Θi ) denote the frequency of a valuation type θin . So φ : {Θni }i,n → {Φni }i,n ,
and the notation φ[θin ](θk ) denotes the fraction of items having a specific value θk .25
When there is no confusion, we also use φni ∈ Φni for i’s true frequency, and φbni for a
feasible frequency that can be announced.
Recall that S(θsn , θbn ) is the surplus with the corresponding pair of valuation types. With
a slight abuse of notation, we extend the definition of this function to capture the expected
surplus as a function of a frequencies:
• S(θsn , φnb ) =
P P
• S(φns , θbn ) =
P P
• S(φns , φnb ) =
P P
k
k
k
θbk
φnb (θbk ) · (θbk − θsk )+
θsk
φns (θsk ) · (θbk − θsk )+
θsk ,θbk
φns (θsk )φnb (θbk ) · (θbk − θsk )+
Note that S(φ[θin ], φnj ) = S(θin , φnj ), ∀θin , φnj , i.e. the expected surplus (given a frequency
of the other agent) depends only on one’s true frequency φ[θin ], due to the independence
across agents’ valuations.
Although the agents’ beliefs are defined over each other’s valuation types θin , when both
agents use exchangeable strategies in Phase 1, the above observation implies that it suffices
to focus on each other’s frequencies (when analyzing beliefs at any nodes except the last part
of Phase 2 at which point beliefs are not longer relevant).
25
Notice that a term in the square brackets is the valuation type, i.e. a (n-)vector, whereas a term in the
parentheses is a number. For instance, if the seller s’s valuations are drawn from {0,8} for each of the n = 5
items, then Φ5s = {(x, y) ∈ {0, 51 , ..., 1}2 | x + y = 1}; and with a type θs5 = (0, 0, 8, 8, 8), s’s true frequency is
φ[θs5 ] = ( 52 , 53 ), where φ[θs5 ](0) = 52 and φ[θs5 ](8) = 35 .
31
We next define sets of frequencies that are less than some pre-specified distance d from the
expected frequency: Φni (d) ≡ φni : |φni − f i | < d and Φn (d) ≡ (φns , φnb ) : |φni − f i | < d, i = s, b ,
where | · | is sup norm. Note that the sets naturally depend on f¯i ’s, but we omit them in the
notation since they are fixed throughout the statement and proof of the theorem.
Let
αi0 (n, d) ≡ Pr(|φni [θi ] − f i | ≥ d)
be the time-0 prior on frequencies that differ by at least d from the expected frequency. Let
αit (n, d, γ) ≡ (γn )−t αi0 (n, d), where γ is the total rate of trembles and γn = maxi |Φγn |·|V n | is
i
the (minimal) rate of trembles to each action when there are n items. Then, by Lemma
1, conditional on any history ht up to time-t, the likelihood of frequencies that are at least
distance d from the expected frequency is bounded above by αit (n, d, γ), i.e.
Pr(|φni [θi ] − f i | ≥ d|ht ) ≤ αit (n, d, γ)
(1)
When there is no confusion, we simplify notation by using αi0 and αit ’s, for a given set of
parameters (n, d, γ).
The proof proceeds as follows: We first show the existence of exchangeable equilibria. Then
we extend the idea in the proof of Theorem 2, providing expected payoff bounds. Due
to the uncertainty about overall surplus, we are no longer able to provide useful bounds
over all types nor in an ex-post sense; however we can focus on the types that are less than
distance d from the expected frequency and bound their expected payoffs. The bounds do not
exactly pin down one’s payoffs, but approximately so - thanks to the bounds on posteriors.
In addition, we show that the overall errors brought by types that are at least distance d
from the expected frequency and approximate bounds vanish as n becomes large. Finally
we illustrate that of the vanishing errors imply our main statements, i.e. the approximate
efficiency and uniqueness of divisions.
Existence of exchangeable equilibria.
For any original bargaining game G with n items, construct an induced game G˜ as follows:
Suppose an agents i only observes i’s frequency type φ[θin ], instead of the valuation type θin ,
until the beginning of Phase 2. (The phase 1 of) such a game G˜ has finite type spaces, finite
action in each period, and “continuity at infinity”, thus has an sequential equilibrium.26
˜ we construct the following exchangeable stratGiven a sequential equilibrium of G,
egy/belief profile for G which is also an equilibrium. In Phase 1, an agent i with type
˜ The belief system
θin copies the type φ[θin ]’s strategy in the sequential equilibrium for G.
26
“Continuity at infinity” means the (time-0) continuation value of the game after period T vanishes as T
goes to infinity. See, Fudenberg and Levine (1983), p.258 for the definition, and p.267 Theorem 6.1 for an
existence result.
32
˜ so that in each information set an agent/type θin shares
induced by the belief system of in G,
the beliefs that the type φ[θin ] has in game G˜ over the frequency space; and over the valuation
type spaces, the beliefs are equally assigned to types corresponding to a same frequency.
The strategy is exchangeable by construction. In addition, such a strategy/belief profile
is a Bayesian equilibrium of the original game G: In Phase 1, for any i, given that the
other agent always assign the same beliefs over i’s types with a same frequency, i cannot be
strictly better off by deviating to an non-exchangeable strategy. Therefore we have shown
the existence of exchangeable Bayesian equilibria in the original game G.
Expected payoff from an offer.
Exchangeability of strategies, together with the iid distributions and exchangeable trembles, implies the following strategies as part of an equilibrium continuation in Phase 2: The
n
n
} and pay an amount that
> θsk
recipient has a unique strict best reply to trade X = {k | θbk
exactly delivers the payoff demanded by the offerer - provided these lead to positive payoffs,
and otherwise to say ‘No’ (and do either if there is indifference),27 If the offerer was truthful
in the first Phase on the announced frequency, then given recipient’s strategy is to truthfully
list valuations. (What happens in other subgames will not be important for the argument
below.)
The Phase 2 strategies imply the following expected payoff from offering/accepting an
offer in Phase 1: An offerer of type θin gets a (non-discounted) payoff of vbi with a “truthful”
offer (φ[θin ], vbi ) if the offer is accepted; recall that φ[θin ] is the true frequency of θin . A recipivi .
ent’s (non-discounted) expected payoff from accepting an offer (φbni , vbi ) is Evj = S(θjn , φbni )−b
This is true regardless of whether φbni is i’s true frequency, since that is the constraint subject
to which i has to list valuations in Phase 2.
Payoff bounds for agents with frequencies that are less than distance d from the
expected frequency.
Next we bound the expected payoffs of agents whose frequencies are less than distance
d from the expected frequency in any Bayesian equilibrium, and then show the upper and
lower bounds to an agent’s payoffs are close to each other. Formally, define the following
payoff bound(s), for i ∈ {s, b}:
• Mi (αs , αb ; d, n) [Li (αs , αb ; d, n)] is the sup [inf] of expected payoff from the continuation
of the game (discounted to the current point of the game) that agent i can obtain in
any equilibrium, with any φni ∈ Φni (d), and at any decision node of the game such that
the current posteriors satisfy Pr(|φns [θs ] − f s | < d|ht ) ≥ 1 − αs and Pr(|φnb [θb ] − f b | <
d|ht ) ≥ 1 − αb .
27
Generally, if items can have the same value for buyers and sellers then the strategy specification on
whether those particular items trade is undetermined and does not influence the argument.
33
When there is no confusion we write them as Mi (αs , αb ) and Li (αs , αb ), but notice that the
payoff bounds do depend on (d, n).
Note that the expected surplus with any type whose frequency is less than distance d
¯
from the expected frequency is close to the limit surplus S:
¯ < 2dSmax , ∀(φn , φn ) ∈ Φn (d).
|ES(φns , φnb ) − S|
s
b
(2)
1. We now show that when i makes an offer at t and j is the responder:
1
(3)
Mi (αst , αbt ) ≤ S¯ + 2dSmax − δj (1 − αjt+1 )Lj (αst+1 , αbt+1 ) +
n
where the αit ’s are the previously defined bounds on posteriors (of frequencies that at least
distance d from the expected frequency).
Li (αst , αbt )
≥ (1 − γ)(1 −
αjt )
1
t+1
t+1
S¯ − 2dSmax − δj Mj (αs , αb ) −
n
(4)
Proof of (3) and (4):
(3) is straightforward, by noting that S¯ +2dSmax is an upper bound on the expected total
surplus that remains by (2), and δj Lj (αst+1 , αbt+1 ) is a lower bound of j’s expected present
value of rejecting i’s current offer, with the extra n1 being the largest possible (per-item) loss
due to the unit gap of payoff grids.
(4): Noting that S¯ + 2dSmax is a lower bound on the expected total surplus that remains
by (2) Consider an offer from i with her true frequency φni and any demanded payoff of
nor more than S − 2dSmax − δMj (αst+1 , αbt+1 ). Such an offer will be accepted by j with
any frequency φnj ∈ Φnj (d), since j’s payoff from this offer exceeds the present value of
Mj (αst+1 , αbt+1 ), the upper bound of what she can get when rejecting the offer. Hence the
probability of acceptance is at least (1 − αjt )(1 − γ), with 1 − γ being the likelihood that
trembles do not apply.
2. Iteratively applying Equations (3) and (4) lead the following time-0 bounds on payoffs
(assuming S is the offerer in the initial period, the other case is analagous).
First let δmax ≡ max{δs , δB }. Next let error2t be a bound on “error terms” that will
bound how far expected payoffs can differ from the Rubinstein shares, which is defined by
error2t = (αs2t +αb2t +δmax (αst+1 +αbt+1 ))Smax +(1+δmax )2dSmax +(1+δmax ) n1 +δmax γSmax .
Then it follows that
PT −1
P2T −1 t
2T −2
1−δmax
1
2t
t
t
t=0 error =
t=0 [δmax (αs + αb )] Smax + 1−δmax (2dSmax + n + δmax γSmax ).
Then, from an iterative application of (3) and (4):
¯ − δb ) + δs δb Ms (α2 , α2 ) + error0
Ms (αs0 , αb0 ) ≤ S(1
s
b
¯ − δb )(1 + δs δb ... + δsT δ T ) + δ T δ T Ms (α2T , α2T ) + PT −1 error2t
= S(1
s b
s
b
b
t=0
PT −1
δsT δbT
1−δb ¯
2t
S+(
Smax +
error ),
≤
1−δs δb
1−δs δb
t=0
34
(5)
¯ − δb ) + δs δb Ls (α2 , α2 ) − error0
Ls (αs0 , αb0 ) ≥ S(1
s
b
P −1
¯
= S(1 − δb )(1 + δs δb ... + δsT δbT ) + δsT δbT Ls (αs2T , αb2T ) − Tt=0
error2t
P
T −1
1−δb ¯
S − t=0
error2t .
≥ 1−δ
s δb
(6)
This implies that
1−δb ¯
S
1−δs δb
−
PT −1
t=0
error2t ≤ Ls (αs0 , αb0 ) ≤ Ms (αs0 , αb0 ) ≤
1−δb ¯
S
1−δs δb
2T
+ (δmax
Smax +
PT −1
t=0
error2t ),
(7)
3. Next, we show that all “error” terms go to 0 as n → ∞. In particular, for ∀ε > 0,
6(1−δs δb )η (1−δs )δb η
η
4η
+ 5Smax
, 5δ
∀δs , δb < 1, first pick η > 0 such that max{4η, δ(S−η)
¯
¯ , 1−δb S¯ } < ε - this
b (1−δs )S
is the tolerance level of errors in payoff bounds that we allow for. Then in turn:
2T
Smax < η/5
• pick T ∈ Z+ such that δmax
• pick d > 0 such that
• pick γ(ε) <
2dSmax
1−δmax
ε
,
5(1−δmax )Smax
< η/5
hence
γ(η)Smax
1−δmax
< η/5
• for any γ ∈ (0, γ(ε)) and d (already picked),
pick the threshold numberi of items, nγ ,
P2T −1 h [ 2t ] [ 2t ]+1 t
(αs (d, n, γ) + αbt (d, n, γ)) Smax < η/5.
so that for ∀n > nγ we have t=0 δs δb
To do so, recall αit (d, n, γ) = γn−t αi0 (d, n), where
γn−t ∼ O(nmaxi |Θi |t ), where |Θi | is the number of feasible valuations (for each item);
2
n
n
0
b
αi = Pr |φi − f i | ≥ d ≤ 2e−2d n (the Dvoretzky-Kiefer-Wolfowitz (1956) inequality),
hence fix any t, αit = γn−t αi0 → 0 as n → ∞; so does their discounted sum (up to
T − 1),
• finally, if
1
nγ (1−δmax )
≥ η/5, replace nγ by
5
η(1−δmax )
so that
1
n(1−δmax )
< η/5 for ∀n > nγ .
4. We now put the pieces together to obtain tight equilibrium payoff bounds. In any
equilibrium, the seller’s expected time-0 payoff
1 − δb ¯
1 − δb ¯
0
n
EUs (φs ) ∈
S − η,
S + η , ∀n > n, ∀φns ∈ Φns (d)
(8)
1 − δs δb
1 − δs δb
Similarly, the buyer’s expected time-1 payoff (whenever time-1 is reached)
1 − δs ¯
1 − δs ¯
1
n
EUb (φb ) ∈
S − η,
S + η , ∀n > n, ∀φnb ∈ Φnb (d)
1 − δs δb
1 − δs δb
Hence the buyer’s expected time-0 payoff, in any equilibrium, is at least
35
(9)
EUb0 (φnb )
≥ δb (1 − γ)
1 − δs
−η
1 − δs δb
>
δb (1 − δs ) 6
− η, ∀n > n, ∀φnb ∈ Φnb (d)
1 − δs δb
5
(10)
By the construction of η, we have EUs0 (φns ) and EUs0 (φns ) are both in the region of (1 −
ε, 1 + ε) times the corresponding Rubinstein shares with the limit surplus.
Realized surplus and likelihood of immediate trade.
From Equations (8) and (10), ∀n > n, in any equilibrium, the realized surplus is at least
EUs0 (φns ) + EUb0 (φnb ) > S¯ − 3η, ∀(φns , φnb ) ∈ Φn (d)
(11)
In expectation, the surplus realized is at least:
Pr(Φ (d))(S¯ − 3η) ≥
n
η
1−
5Smax
(S¯ − 3η) > S¯ − 4η > S¯ − ε
recall Pr(Φn (d)) = (1 − αs0 )(1 − αb0 ) ≥ 1 − 2d >
η
5Smax
(12)
for ∀n ≥ n.
Now we turn to the likelihood of immediate trade:
With any pair of types whose frequencies are less than distance d from the expected
frequency, the maximal surplus is at most S¯ + η, and the total cost of delay for one period
is at least δ(S¯ − η). Hence with such types, the likelihood of delay is at most Pr(delay |
ΦnEmp (d)) = 4η/ δ(S¯ − η) . This gives a bound on overall delay:
Pr(delay) ≤ Pr(delay | Φn (d)) Pr(Φn (d)) + 1 − Pr(Φn (d))
4η
η
≤ ¯
+
δ(S − η) 5Smax
<ε
36
(13)
Supplementary Appendix: Additional Results with a
Nearly-Known Surplus
(to “The Efficiency of Bargaining with Many Items”
by Matthew O. Jackson, Hugo F. Sonnenschein, and
Yiqing Xing)
Subsection 5.2 discussed a (technical) challenge associated with the multiplicity of sequential
equilibria due to dramatic updating in posteriors. This challenge was handled in subsection
5.3 by introducing trembles and working with the frequency protocol. Here, we present
two other approaches. The first approach imposes a restriction on how fast beliefs can be
updated. The second approach considers trembles, but with a fixed number of objects.
Thus, in the second approach the convergence of beliefs is not derived from the law of large
numbers but must come from some justification based on the knowledge the agents have
about their environment.
The main advantage of these approaches is that they work with all protocols that have
share-demanding offers (see definition 1, including our first protocol from Section 2.3.1 and
the combinatorial protocol from Section 2.3.3, as well as their variations that allow for
continued bargaining over not-yet-traded objects).
Both approaches consider a sequence of bargaining problems {(nm , Θ, f m )}m=1,2,... whose
(per-object) surpluses converges to S¯ > 0, i.e.
¯ as m → ∞
S m →p S,
where recall that S m is the per-item surplus in the m-th problem, i.e. a random variable
P
S(θs , θb ) ≡ n1m k=1,...,nm (θbk − θsk )+ with types (θs , θb ) drawn according to f m .
We do not impose further additional assumptions, hence allow for, for instance:
• correlations across players (e.g., common shocks),
• correlations across items, and
• asymmetries across items.
We use m as an index since the numbering of the sequence may differ from the number
of objects. For example, the second approach has nm = n, ∀m. Thus, the sequence applies
to settings in which agents have increasingly accurate knowledge of the surplus based on
some fundamental economic reason - e.g., having good information about the environment rather than just relying on the law of large numbers.
1
Let the m-th problem have a surplus grid V nm ,∆ = {0, ∆, 2∆, ..., nm Smax } from which an
agent can demand a total surplus.
A.1
Approximate Efficiency Results with Bounded Belief Updating
For simplicity we consider protocols for which all agents’ past actions are commonly observed; e.g., our first protocol from Section 2.3.1 and the combinatorial protocol from Section 2.3.3. In such protocols, at the beginning any period t agents share a common history
t−1
, at−1
ht−1 ≡ (a1i(1) , a1j(1) , . . . , ai(t−1)
j(t−1) ), and after the offerer moves the common history bet−1 t
comes (h , ai(t) ). We denote the set of all possible histories by H, including h0 ≡ ∅ being
the initial decision node of the bargaining game.
An agent i’s beliefs fej : H → ∆(Θnj ) map histories to a distribution over the other agent’s
type space. In particular, we let fej (E, ht−1 , θi ) denote i’s belief over E conditional history
ht−1 . Note that an agent i’s posterior belief can depend on i’s own type θi .
The beliefs at the initial node (before types are drawn) are the common prior, i.e. fe(· |
∅) = f (·).
We require agents’ initial beliefs conditional upon their types to be consistent with the
common prior, in particular, fej (·, ∅, θi ) = fj (·, θi ), ∀θi , where fj (·, θi ) is the marginal distribution (of f ) over Θnj .
Now we introduce a restriction on how fast beliefs can be updated. We say a belief system
fej (·) satisfies bounded updating at rate β ≥ 1 if for ∀ E ⊂ Θn , ht−1 ∈ H, ati(t) ∈ Ai(t) (ht−1 ),
and θi ∈ Θni :
fej (E, (ht−1 , ati(t) ), θi ) ≤ β fe(E, ht−1 , θi );
and for ∀ E ⊂ Θn , (ht−1 , ati(t) ) ∈ H, atj(t) ∈ Ai(t) (ht−1 , ati(t) ), and θi ∈ Θni :
fej (E, (ht−1 , ati(t) , atj(t) ), θi ) ≤ β fej (E, ht−1 , ati(t) , θi ).
We consider (arbitrarily) large but bounded βs. This means that the assumption only
affects events that are very unlikely according to the prior. In particular, when fe(E | ht−1 ) >
1
, the constraint is not binding for the updating at history ht−1 .
β
Next, we introduce an equilibrium notion with bounded belief updating.
Definition 2 (Equilibrium with bounded belief updating) An equilibrium with bounded
belief updating at rate β is a profile of the agents’ (mixed) strategies and posterior systems
fej , j = s, b, such that
1. At any decision node, the mover i’s strategy maximizes his/her expected payoff given
the other’s strategies and his/her posterior system about the other’s types fej ;
2. Both agents’ posterior systems fes and feb satisfy bounded updating at rate β.
2
Definition 2 imposes only minimal requirements on beliefs other than the boundedupdating requirement. In particular, we do not require that agents’ posterior systems be
induced by some joint posterior system, nor do we require that the agents’ posterior systems
are common knowledge, nor do they even have to be consistent with Bayes’ rule. So this can
be viewed as a notion that allows for the most possible outcomes as equilibria, under some
(arbitrarily large) bound on updating. We show that even with such a minimal restriction,
all equilibria are approximately efficient with vanishing uncertainty about overall surplus.28
Theorem 4 Consider a sequence of bargaining problems {(nm , Θ, f m )}m=1,2,... such that the
distributions {f m } have a converging per-item surplus S¯ > 0, and the protocol includes sharedemanding offers. For any ε > 0, ∀β ∈ [1, ∞), ∀δs , δb < 1, there is ∆(ε) > 0 such that for
any ∆ ∈ (0, ∆(ε)) there exists m∆ such that if m > m∆ then:
1. There exist equilibria with bounded belief updating at rate β;
2. In any such equilibrium, with probability at least 1 − ε:
• agreement is reached in the initial period;
¯ and
• the realized surplus is at least (1 − ε)S;
• Expected payoff / ‘Full-Information Rubinstein share for the realized types’ for
each agent lies in (1 − ε, 1 + ε).
A sketch of the proof of Theorem 4:
Most parts of the proof are similar to Proof of Theorem 3 and so are not repeated here. The
new feature of the current theorem is that it does not impose the assumption of independence
across agents’ types, so that an agent’s beliefs about the other’s types may depend on his
or her own type. This brings extra steps in proving the theorem, which are our main focus
here.
¯ < d} be
First, for any distance d > 0 and any type θi , let Θnj m (θi , d) = {θj | |S(θi , θj ) − S|
the set of other’s types for which the per-item surplus is close enough to the limit S¯ (within
a distance of d).
Recursively, construct the following sequences of subsets of the agents’ types (for i = s, b),
given some ι0 , ι1 , ... > 0
e nm (d, 0, ι0 ) = θi | fb Θnm (θi , d), θi > 1 − ι0 , i = s, b;
0. Θ
i
j
n
o
e nm (d, 1, ι1 ) = θi | fb Θnm (θi , d) T Θ
e nm (d, 0, ι0 ), θi > 1 − ι1 , i = s, b;
1. Θ
i
j
j
28
An alternative notion would apply a variation on sequential equilibria, but with the modification that
“beliefs are rounded to the boundary”; i.e., if the posterior on any event exceeds β times the prior, that
belief on that event is replaced by β times the prior. Same theorem holds under that alternative notion.
3
...
n
o
e nm (d, t, ιt ) = θi | fb Θnm (θi , d) T Θ
e nm (d, t − 1, ιt−1 ), θi > 1 − ιt , i = s, b;
t. Θ
i
j
j
...
Intuitively, for an agent i with any type in the t-th set above, i’s prior is such that with
a probability of at least 1 − ιt the surplus is close to the limit S¯ (within a distance of d) and
that the other has a type in the (t − 1)-th set. In terms of posteriors: for any period t0 , i’s
0
posterior (following any history up to that period) is at least β t (1 − ιt ) for the above events.
To simply notation we omit the superscript nm when there is no confusion.
We provide payoff bounds for those sets. In particular, for some fixed T ∈ N and
{ι0 , . . . , ι2T } (we discuss how to pick these below), for i ∈ {s, b}, let Mit [Lti ] be the sup [inf]
of the expected payoff from the continuation of the game (discounted to the current point
e i (d, 2T − t, ι2T −t ).
of the game) that agent i can obtain in any equilibrium, with any θi ∈ Θ
We derive the bounds recursively:
• Backward from period 2T :
Mi2T < β 2T (1 − ι2T −t )(1 − ι0 )(S¯ + d) + (1 − β 2T (1 − ι0 ))Smax ; and
L2T
i ≥ 0.
...
• In period t < 2T , agent it makes the offer:
t
t
< β t (1 − ι2T −t )(S¯ + d − δi(t+1) Lt+1
Mi(t)
i(t+1) ) + (1 − β (1 − ι2T −t ))ι2T −t Smax , which is the
maximum surplus left minus the share that must be delivered to the other agent with
e i(t+1) (d, 2T − t, ι2T −t ); and
types in Θ
t+1
Lti(t) > (1 − β 2T −t ι2T −t )(S¯ − d − δi(t+1) Mi(t+1)
− ∆), since any offer that delivers at least
2T
e i(t+1) (d, 2T − t, ι2T −t ).
δs Ms is accepted for sure by the other agent with types in Θ
...
e i (d, 2T, ι2 T ). It is easy to
The above process bounds time-0 payoffs for the types in sets Θ
verify that (for any fixed T ) when d, ι0 , ι1 , ..., ι2T > 0 and ∆ > 0 go to 0, the time-0 bounds
Ms0 and L0s become arbitrarily close to each other, hence the expected payoff for any type in
e 2T, ι2T ) is approximately determined. In addition, that payoff can be arbitrarily close
Θ(d,
¯ for large enough T .
to the corresponding Rubinstein share with a surplus S,
The final step is to show that the above bounds have bite for most types (according to
prior probabilities). Formally, notice that (fixing any T ) for large enough m, i.e. as the prior
4
knowledge over surplus
becomesprecise enough, we can find small enough d, ι0 , ι1 , ..., ι2T > 0
e
while keeping Pr Θi (d, 2T, ι2T ) close enough to 1.
¯ < d) which converges to 0 according to the
To do so, for any d > 0, let α = Pr(|S − S|
definition of converging surplus. We construct ι’s from α0 :
√
e i (d, 0, ι0 )) ≥ √α = ι0 ,
• ι0 = α, easy to verify that fi (Θ
α
¯ < d and θj ∈ Θ
e j (d, 0, ι0 ) ≥ 1 − (α + ι0 );
hence Pr |S − S|
• ι1 =
√
e i (d, 0, ι0 )) ≥
α + ι0 , easy to verify that fi (Θ
√α+ι0
α+ι0
= ι1 ; and
...
• ιt =
√
α + ιt−1 ;
...
Continue this process until we get ι2T , which converges to 0 as α goes to 0 (i.e. with large
enough m).
e i (d, 2T, ι2T ),
In summary, we can approximately determine expected payoffs for all types in Θ
whose (prior) probability is at least 1 − ι2T , for arbitrarily small ι2T as m becomes large.
The rest of the proof parallels the corresponding parts of the proof of Theorem 3.
A.2
Trembles with Fixed Number of Objects.
When there is no confusion we write Mi (αs , αb ) and Li (αs , αb ), but note that the payoff
bounds depend on (d, n). We now work with sequences of economies, where the number(s)
of objects are bounded above (or fixed), and in which uncertainty over total surplus vanishes
but substantial uncertainty about each object remains. This captures agents have accurate
information about the surplus rather than relying on laws of large numbers to give them
accurate information about the surplus.
The bound on the number of objects implies boundedness of sizes of action spaces (given
any increment ∆ > 0 in the grids of surplus), and thus allows for an approximate efficiency
results derived with trembles, instead of putting an artificial restriction on beliefs.
A bounded number of objects and the vanishing uncertainty over overall surplus need
not contradict with each other: the assumption captures an environment in which agent’s
knowledge of each other’s total valuation is strong, which is natural in many settings.
In particular, we consider a sequence of bargaining problems {(n, Θ, f m )}m=1,2,... (note
the additional restriction that nm = n, ∀m), and again we work on the finite grids surplus
V n,∆ . Trembles are introduced similar to those in subsection 5.3: consider trembles in any
period by all types of any agent with probability γ, uniformly to each of the feasible actions.
5
Theorem 5 Consider a sequence of bargaining problems {(n, Θ, f m )}m=1,2,... such that the
distributions {f m } have a converging per-item surplus S¯ > 0, and the protocol includes sharedemanding offers. For any ε > 0, there exist a small enough tremble probability γ(ε) > 0 and
increment of grids ∆(ε) > 0 such that for any γ ∈ (0, γ(ε)) and ∆ ∈ (0, γ(ε)) there exists
mγ,∆ such that if m > mγ,∆ then:
1. There exist (perfect) Bayesian equilibria with trembles;
2. In any such equilibrium, with probability at least 1 − ε:
• agreement is reached in the initial period;
¯ and
• the realized surplus is at least (1 − ε)S;
• Expected payoff / ‘Full-Information Rubinstein share for the realized types’ for
each agent lies in (1 − ε, 1 + ε).
We omit a formal proof of Theorem 5. The existence part is straightforward. The rest
of the theorem follows by the same logic as Theorem 4, since with fixed number of items
(hence fixed action space) and trembles, beliefs are updated at bounded rates (per-period)
that are invariant to m. The only exception involves the errors directly due to the trembles,
which are arbitrarily small as the total size of trembles goes to 0.
6